Student Probe Graphs of Proportional Relationships Susan runs three laps at the track in 12 minutes. A graph of this proportional relationship is shown below. Explain the meaning of points A (0,0), B (1,4), and C (2,8) in terms of this relationship. Answer: A—Susan runs 0 laps in 0 minutes. This is the y-­‐
intercept. B—Susan runs 1 lap in 4 minutes. This is the same as of a lap in one minute (unit rate or constant of proportionality). C—Susan runs 2 laps in 8 minutes. Lesson Description This lesson expands upon students’ understanding of ratio tables and unit rates to graph and interpret proportional relationships. If students have difficulty with this lesson more time should be spent on the prerequisite lessons Ratios and Proportional Thinking, Unit Rates, and Equations of Proportional Relationships. At a Glance What: Describe the graph of a proportional relationship in terms of the situation Common Core State Standard: CC.7.RP.2d Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Matched Arkansas Standard: AR.8.A.6.3 (A.6.8.3) Algebraic Models and Relationships: Differentiate between independent/dependent variables given a linear relationship in context AR.8.A.7.1 (A.7.8.1) Analyze Change: Use, with and without technology, graphs of real life situations to describe the relationships and analyze change including graphs of change (cost per minute) and graphs of accumulation (total cost) Mathematical Practices: Reason abstractly and quantitatively. Model with mathematics. Use appropriate tools strategically. Who: Students who cannot describe the graph of a proportional relationship in terms of the situation Grade Level: 7 Prerequisite Vocabulary: ratio, proportion, ratio table, unit rate Prerequisite Skills: graphing on the coordinate plane, finding unit rates, creating ratio tables Delivery Format: Individual, small group Lesson Length: 30 minutes Materials, Resources, Technology: graph paper, graphing calculator (optional), straight edge Student Worksheets: Coordinate Grid Paper Rationale This lesson builds upon students’ understanding of proportional relationships to begin their study of functions. As students become more adept at graphing and interpreting these relationships, they are building the foundation for graphing more general linear functions, and then all functions. Preparation Provide straight edges and several copies of Coordinate Grid Paper for each student. Lesson The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 1. Complete a ratio table for Refer to Ratio Tables. Spanish All this proportional Students Students relationship: 1 3 One out of three students 2 6 at Central Middle School 3 9 take Spanish. 4 12 5 15 2. What is the unit rate of 3. Refer to Unit Rates. this relationship? There are 3 students in the How do you know? school for each Spanish student. 3. How many students are in 0 the school if 0 students take Spanish? 4. Add a column to your Prompt students. Spanish All Ordered ratio table and write the Refer to Graph Ordered Pairs Students Students Pairs numbers as ordered pairs. on a Coordinate Plane. 1 3 (1,3) 2 6 (2,6) Let’s graph the ordered 3 9 (3,9) pairs on your grid paper. 4 12 (4,12) 5 15 (5,15) 5. What does the ordered If 5 students take Spanish, What do the numbers on the pair (5,15) tell us? there are 15 students in the x-­‐axis represent? school. What do the numbers on the y-­‐axis represent? The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 6. Using your straight edge, Students’ lines should go Monitor students. Model, if draw a line through all the through the origin. necessary. points. Notice that the line goes If there are no students taking through the point (0,0). Spanish, then there are no What does that mean? students in the school. 7. Another important point is For every 3 students in the Remind students that the unit (1,3). school, there is one student rate was 3. What does this ordered taking Spanish. pair mean? What was important 3 is the unit rate. about 3 in this problem? 8. Since the line goes through the origin, we say the y-­‐intercept (where the line crosses the y-­‐axis) is 0. Since the unit rate of the relationship is 3, we say 3 is the slope of the line. We can graph any line if we know its slope and y-­‐
intercept. 9. Let’s use what we have Refer to Equations of learned to investigate Proportional Relationships. another proportional relationship: Tracy can ride her bike 30 miles in 3 hours. 10 miles per hour. What is the unit rate? In one hour she can bike 10 How do you know? miles. 10. So that gives us the ordered pair (1,10). 11. If she bikes for 0 hours, 0 miles how far did she travel? What ordered pair does (0,0) that give us? 12. Notice that we have the slope (the unit rate is 10) and the y-­‐intercept (0). The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 13. Now we have two Students should graph the Monitor students. Model, if ordered pairs to plot. Plot line with a slope of 10 and necessary. them on grid paper and passing through the origin. draw a line through them with your straight edge. 14. Pick another point on the Answers will vary. For line. What does this point example, the point (4, 40) mean? means that Tracy bikes 40 miles in 4 hours. 15. Repeat with additional problems as necessary. Teacher Notes 1. This lesson is a good vehicle for introducing the terms slope and y-­‐intercept if your students are not already familiar with them. 2. It is important for students to make the connection between unit rate (constant of proportionality) and slope. 3. The concept of independent and dependent variables may need to be revisited during this lesson. Variations This lesson may be adapted to graph equations of proportional relationships on the graphing calculator and interpret the points using technology. Formative Assessment Jill bought a box of 12 cupcakes for $4.80. A graph of this proportional relationship is shown below. Explain the meaning of points A (0, $0), B (1, $0.40), and C (5, $2.00) in terms of this relationship. Answer: A—Jill paid $0 for 0 cupcakes. The y-­‐intercept is 0 B—Cupcakes cost $0.40 each. This is the unit rate and the slope of the line. C—Five cupcakes cost $2.00. References
Greenes, C. (2000). Groundworks to Algebraic Thinking. Columbus, OH: Wright McGraw Hill. Russell Gersten, P. (n.d.). RTI and Mathematics IES Practice Guide -­‐ Response to Intervention in Mathematics. Retrieved December 7, 2011, from rti4sucess: http://www.rti4success.org/images/stories/webinar/rti_and_mathematics_webinar_presentati
on.pdf Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on December 7, 2011 from http://iris.peabody.vanderbilt.edu/case_studies/ICS-­‐010.pdf Van de Walle, J. A., & Lovin, L. H. (2006). Teaching Student-­‐Centered Mathematics Grades 5-­‐8 Volume 3. Boston, MA: Pearson Education, Inc.